It has been said, in many publications and news reports, that there was a
period of time that the launch codes for Minute Man nuclear missiles was
00000000. This was because those in the military were responding to
President Kennedy's mandate that an extra layer of security (or, if you
will, an extra mechanism) be added to the launch procedures for a nuclear
missile. Those in charge of such things in the military, however, were
more concerned with the ability to launch a missile and considered that
extra step a hindrance. Because of that, they did in fact implement that
extra step, but the extra step required an eight-digit code which they
simply set to 00000000.

A simple search of the internet can find this rumor on many sites. The
government has only recently DENIED that the launch codes were ever set to
all zeroes.

Because no one would EVER try 00000000 first before trying 00000001 all the way to 99999999.

I thought that was the point. According to the UL, the military was required to add the extra "security" step, but since they prioritized a quick launch process, they deliberately chose the simplest (and correspondingly least secure) code possible. The code wasn't chosen for security (i.e., because it was unlikely to be guessed), it was chosen to be easy to remember and punch in--and 00000000 fulfills that purpose quite well.